His wanted Holden Colorado was spotted by a member of the public about midday on October 10, sparking a police operation involving special officers and a helicopter.
A body, later identified to be Greig, was found inside the burnt-out Holden in bushland in the afternoon.
An inquest, which began in Hobart today, is examining the circumstances and cause of his death.
Counsel assisting the coroner Cameron Lee said Greig had been in trouble with the law from an early age and had served jail time after evading police.
On the night of September 26, Greig went to the home of a woman who had taken out a family violence order against him, Lee said during opening submissions.
He hit her five to six times in the head, saying “this is your fault” and “till death do us part”.
Police were alerted and as they rounded a corner near the house they heard multiple “pops” and a window of their car shattered.
Lee said eight to 10 shots were fired at the car, with one officer noting they came from behind a hedge in a paddock.
An officer in a different car who heard a radio transmission about the shooting was confronted by a 4WD driving with its high beam and roof lights on at nearby New Norfolk.
When the car was about 50m away, the officer heard about five gunshots.
“It was at this point the search for Greig was on,” Lee said.
Greig set up camp in bushland not far from New Norfolk, where his was routinely picked up by his partner at 7.30pm, Lee said.
The pair, who communicated by letters left under a wheelie bin, often spent the night at her house.
Greig’s dad Gary told the inquest he believed police had shot and killed his son and burned the car to cover evidence.
Gary said a lawyer told him a source said police had used a flash grenade – but the lawyer later denied making the statement.
Under cross examination, Gary admitted evidence indicated his son wasn’t shot by police and there was no proof officers discharged their guns.
“I just want to know the last 10 minutes of his life,” he said.
Greig had talked to his partner about getting 10-20 years in jail for the shootings, while concerns had previously been raised about his mental health, Lee said.
Guns were found in Greig’s car and Lee said two specialist officers would give evidence of hearing a single shot.
He said investigations deemed the blaze deliberate and a post-mortem examination was unable to conclusively determine a cause of death because of the fire.
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